Limit theorems for a triangular scheme of U-statistics with applications to inter-point distances (Q1083110)

Let \(X_ 1,X_ 2,..\). be a sequence of independent and identically distributed random variables and for each \(n=2,3,..\). let \(f_ n(x,y)\) be a measurable, symmetric function of two variables. This paper provides conditions under which the statistic \[ U_ n=\sum_{1\leq i<j\leq n}f_ n(X_ i,X_ j), \] suitably normed, converges in distribution to an infinitely divisible law. Convergence to a normal distribution is considered in some detail; the authors present results similar to those of \textit{N. C. Weber} [Math. Proc. Camb. Philos. Soc. 94, 307-313 (1983; Zbl 0563.60025)] but under different conditions. Applications to interpoint distance statistics are discussed.